Proposition 3.6.3.label Let $\curl{X_i}_{i\in I}$ be non-empty topological spaces, then $X=\prods{}{i\in I}X_{i}$ is Hausdorff if and only if $X_{i}$ is Hausdorff for each $i\in I$.
Proof. Suppose $X_{i}$ is Hausdorff for each $i\in I$. Let $x\neq y\in X$ then $\pi_{i}(x)\neq \pi_{i}(y)$ for some $i\in i$ so $X$ is Hausdorff by Proposition 3.6.2(1). Conversely suppose $X$ is Hausdorff. Since the canonical injection $X_{i}\to X$ is a homeomoprhism for each $i\in I$, $X_{i}$ is Hausdorff by Proposition 3.6.2(2).$\square$
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